Unless the committee is specifically in closed session, the public may walk into any committee hearing at any time, no charge or fee needed. Congress is "the People's House" and part of the set up government is that citizens may freely walk the halls and attempt to talk to politicians who were making their rounds. Speaking as a political nerd, it's kind of fun to be walking the halls of Congress and have an famous senator or representative walk past you (I geeked out when I realized a politician I love walked past me on an escalator. Also, politicians have a pin that they all wear that identifies them as a politician as opposed to a staffer or a lobbyist of press).
The only tickets issued are for the floors of Congress, which can be denied if there is sufficient reason to believe there is going to be some major cause to the gallery having a full attendance (in both houses, the gallery seating runs around the perimeter of the floor, so on a regular day, you can get in without much hassle... don't just show up to attend the State of the Union without an invite.). Typically, you are not allowed on the floor unless invited by a member (Even the President cannot waltz onto the floor during the State of the Union... He must be invited, which is why someone will address the Speaker of the House that the President has arrived before he enters the room to give his address.). You are not allowed to speak or engage in any intentional outbursts during a hearing or while in the gallery and you will be escorted out of the room if you become disruptive. The Gallery has capitol police on guard at all times just to pull out someone who is being disruptive and committees typically have a few by the doors as well.
As there is no assigned seating for the remaining spectator seats, if a major committee is hearing testimony, there may be a line with first come first serve seating enforced. The police will keep an eye on when people leave and new people enter. Those wearing articles of clothing that are disruptive (such as political slogans that are noticeable at a distance) may be barred from entry, but beyond that, they need good reason to keep you out. What is likely going on in the SCOTUS issue is that protesters have entered the line and mixed with people who actually want to watch the hearing in person. When one is disruptive, they are taken out and the next person is permitted entry... and if they too are a protestor, lather rinse and repeat.
If you cannot go to Washington just to sit in committees all day, fear not, as C-SPAN records all business before Congress including committees not closed to the public and will have them online within the day for public consumption. You'll also notice (At least among the Representatives, I've yet to sit in on a Senate Committee) that unless it's a very newsworthy hearing, Reps will not sit for the full committee unless they are the Chair or the Opposition leader of the committee... typically they arrive before their time for questioning and quickly depart, but there are always a few on both sides at any given moment. They do have a busy day and don't always have the time to listen to the full testimony... they have staffers to digest it for them if there are any issues raised when they leave.
To answer a follow up question that posed, getting into Congress building itself is not terribly hard. Security is pretty typical of any government building (think any U.S. Court House) and will require passing items through an x-ray machine and walking through a metal detector and possibly an additional magnetic wand. Visitors will be given a visitor clip tag or sticker to denote who they are. I would recommend you check the official congress website for more details as the last time I personally went in, I was a guest of a congressional staffer and not some guy who walked in off the street. I would also recommend you contact your Representative's and Senators' office about a potential visit as they will typically be helpful... your Representative is the better option as their office is typically not as busy as a Senator's office and it's easier for them to make appointments to actually meet with you.