How political parties view things is not necessarily commensurate with reality. After all, they have a political agenda to push. In this case the view of (federal) judges—Supreme Court Justices or otherwise—as dyed-in-the-wool partisans is not really borne out by the reality. The type of 5-4 split decisions that attract so much attention are in fact a small fraction of the decisions made by SCOTUS, most of which are unanimous or have only one or two dissents, and odd bedfellows of so-called "super liberal" Justices joining with "super conservatives". And it tends to be notoriously difficult to predict how a federal judge, a Justice in particular, is going to rule on things after their appointment.
Justice David Souter was nominated by a Pro-Life Republican President (the first George Bush), was expected to be a solid conservative, and ended up being seen as one of the most liberal, Pro-Choice justices in recent history. And recently we had Justice Gorsuch, who had been touted by Trump as a sure thing for Conservatives, joining a ruling that sex discrimination statutes covered sexual orientation/identification, with the "but-for" test. Which is not to say that Gorsuch hasn't shown an apparent conservative-friendly lean on many cases, but it nevertheless was something of a surprise that the assumed 5-4 "conservative majority" at the time failed to coalesce on an issue that seemed so definitively a Conservative versus Liberal one.
Some tout this as a positive feature of the lifetime appointments all federal judges (including SCOTUS Justices) receive, arguing that because they are freed of political and career pressures they can simply follow and apply the law and constitution in a pure, unobstructed way, unfettered by consequences.
What Justices and (most) Judges do have is a judicial philosophy: a view of how the Constitution and laws are to be interpreted and applied. And certain judicial philosophies are more amenable to conservative or liberal politics. "Originalism" is, pretty much by definitions, quite agreeable to conservative politics, as it basically says "Let's keep doing things the way we used to and have already been doing them". And this sort of judicial philosophy became heavily popularized by the late Justice Scalia, and Trump appointees to SCOTUS have all had strong connections to this philosophy.
Historically speaking, SCOTUS has pretty much always been viewed by Americans as the most apolitical, trustworthy, and ultimately-on-their-side-of-things Governmental institution or branch. Which is a little odd in some ways, because most attempts to measure how "liberal" or "conservative" a given SCOTUS is tends to come to the conclusion that most of them are heavily conservative, and the current Roberts court (pre Scalia's death) was actually a bit left of the historical norm, but still notably conservative. Modern perceptions are perhaps twisted by the lingering memories of the Warren court from the 50's and 60's, which is often considered the most liberal court in US history.
But the entanglement of SCOTUS with politics was almost immediate. Early in the history the court seemed deeply wary of such entanglements, and mostly avoided doing anything significant that might rock the boat, for fear of endangering the young nation. President Washington tried almost immediately to get the Justices to offer him advisory legal opinions on potential laws and actions, and he was kindly turned down (on the basis of the "Cases or Controversies" clause, which the court considered to bar them from such opinions).
Marbury v. Madison, in 1803, was the first time the court ruled on whether acts and actions of the other branches were "constitutional" or not, and in particular asserted that such determinations were a power belonging to it, and they could use it to nullify acts of Congress as well as compel certain members of the Executive to perform certain actions. Now this is quite significant, and the ruling ultimately a masterpiece in retaining that evasive "don't rock the boat" directive despite it being a ground breaking ruling.
The evasion bit arises because, while it declared that a certain act of Congress was unconstitutional, and that the court had the power to compel certain members of the Executive branch to do things, it did not actually do either of those things. The law in question had already been repealed by Congress (though it was in force at the time of the issues at hand in the case), and the court used this Constitutional invalidity to declare itself lacking jurisdiction on the case, and so didn't order anyone to do anything on said technicality (but it did go out of its way to assert that it could, otherwise).
And all of this was necessary because then-President Jefferson was very adversarial toward the court (as was Congress), and was of the opinion that the decision of what is and is not constitutional belonged to him, the President. A ruling which attempted to order him to do anything was, in all likelihood, going to be ignored, and we'd enter into a major constitutional crisis. By declaring they had these powers, but using a (somewhat contrived) technicality to avoid actually using them, SCOTUS robbed Jefferson and Congress of anything substantive to actually gripe about. For in fact Jefferson got what he wanted—to not be told by the court what he/his subordinates must do, especially when it concerned an act from the end of the preceding President's term—and Congress got what they wanted—to not have the courts invalidate one of their laws without their involvement. And SCOTUS got to walk away with powers that not everyone agreed they had before that point.
For a more modern flash point, there is really no more important case to look at than Roe v. Wade, which occurred in 1973, under the Burger court that followed the Warren court. This decision took an issue—abortion—that had to that point been a strictly local, state-level one, and transformed it into a national one. Now the only way that those who disagreed with the ruling could do anything about it was to operate on the national stage, and they would have to go through one of two very difficult routes: amend the constitution or somehow reshape the court and get it to change its mind. National politics was dramatically changed as a result. And ever since, every single SCOTUS nominee has lived under the shadow of that ruling, and every hearing infected by it: Liberals want to know if a Justice will uphold it, and Conservatives want to know if a Justice will overturn it. Even Justices that got confirmed with massive majorities, like Ginsburg and Sotomayor, were grilled on their Roe v. Wade stance. They all generally avoided it. That the court, while often rolling it back inch-wise, never overturned it, despite acquisitions of Republican appointed Justices (such as Souter and O'Connor) and apparent "Conservative majorities", has become a major frustration to many conservatives. While there is more to Conservative vs. Liberal politics than just the issue of abortion, it's unquestionably a major force, and that Roe v. Wade resulted in a massive, national-scale transformation of politics, and overall outlook on SCOTUS.