See About Euclid's Elements and modern video games for one way to understand what doing proofs is supposed to be for. A proof doesn't have to be in words. It's just a situation where knowing one thing is true makes something else true.
For example, a logical syllogism is a proof:
Socrates is a man.
All men are mortal.
Therefore Socrates is mortal.
If you assume the first two lines are true, called the propositions, you know the third line is true. So the final line, the conclusion, is proved by the propositions. But if you wanted to make this proof more broad or rigorous, you could prove one of the axioms from simpler assumptions:
All animals are mortal.
All men are animals.
Therefore all men are mortal.
And then deal with the problem of how you know when something is mortal to make this an even stronger statement. Learn what the formal logical fallacies are to get a sense of what it means to be able to conclude something from previous information. Essentially, there are no rules for what you can't do, but rules for what you CAN do and you have to pretend you can only draw conclusions by moving in the directions allowed by the rules, or else it will break because you made up a new rule that you haven't proven is a rule from the other rules or didn't explicitly say you were taking as axiom.
So the fundamental theorem of calculus you proved from some facts about integrals you might have been given, or might have obtained from facts about derivatives and the definition of an antiderivative. Since the definition is simply whatever the thing is that does what I want it to do, you have to verify this thing is well-defined, unlike "the integer whose square is 1." If you proved that every different integer has a different square, it would make the square root well-defined for every integer which has a square root. This is another form of proof, proof by contradiction: "every different integer has a different square" is true implies "every number with a defined square root has a well-defined square root" is true. So if the first statement is true, the second HAS to be. So you know that if the second's false the first CAN'T be true.
Anyway, if you read some proof found on some Wikipedia page's references from something you're interested in, it will be clearer. You can also read the first few pages of N. Bourbaki's "Algebra 1-3," not understand it for a month, and then go back to reading papers or Wikipedia and suddenly understand everything about mathematics and how it works.