I suspect there is a misunderstanding of what "independent" means. Literally independence means that the branch is not influenced by other branches - this is the spirit of the word used in the question. However, that is not what we typically mean when discussing the independence of a branch of government.
What is independence?
As noted in the question's tags, the idea of independence is closely related to the separation of powers (and checks and balances). The best source for reading about this is Montesquieu's "Spirit of the Laws". If you don't want the primary source, consider at least reading about it on SEP.
A branch is independent when it's functions are performed independently. For the judiciary, it is independent when it is capable of deciding cases without regards to the will of either the executive or legislative branch. Obviously both branches have input into the judicial process (the executive through law enforcement and the legislative through writing laws), but the actual decision is only a matter for the judiciary.
This definition makes sense when put in the context of Montesquieu's work. His concern (shared by many influential people, including many of the Founding Fathers) was that the people need to be protected from the government. The best way to accomplish this is to separate powers into discrete units - "branches". Each branch is interrelated - they have to be in order for government to function - but they are each empowered with unique roles.
How can the judiciary branch be independent while still being interdependent of the other two serving as check and balance?
The judiciary is independent because it alone is able to make binding interpretations of law. Obviously, it has inputs from the other branches, but those inputs don't infringe on its ability to independently come to a conclusion.
This can be contrasted with some other countries where judiciaries are frequently subject to pressure from the executive branch to rule in a certain direction. This pressure is sometimes formal (in that a judge can be removed from office if they rule against the executive branch's wishes) or informal.