There are three levels to understanding contracts.
1) The language itself.
Contracts are written in English (at least in the U.S.), so anyone with a strong understanding of English can read one top to bottom and comprehend what it is saying.
Experience in analyzing writing (like from a few college English courses) can help a lot, both in learning to quickly parse sentences, and in learning to infer motivations from the language on the page. You always want to be able to answer "why did they write it this way? What are they trying to get?"
2) What is typical and standard in contracts.
You can only get this from experience--from reading a lot of contracts--because it's often industry-specific.
For example, many software contracts, even 6 figure enterprise deals, include an ALL CAPS DISCLAIMER OF LIABILITY. THIS SOFTWARE IS OFFERED AS-IS AND WITHOUT WARRANTY. WE DON'T GUARANTEE THIS WILL WORK FOR YOUR PURPOSES, OR INDEED, WORK AT ALL.
I'm exaggerating, but not by much. This is a totally standard clause in the software industry, so whether or not you like it, you generally just have to swallow it if you want to do the deal
Likewise, most EULAs are very similar to one another. Once you've carefully read a few of them, you can quickly skim for the important bits (copyright, liability, data sharing) on new ones.
3) What the law says.
Contracts can't break the law. For example a contract that contains lies would be fraudulent. A contract that says you need to speed or drive drunk is not enforceable. A contract that says you can commit murder would not protect you from prosecution.
This is probably the hardest area to build knowledge, without going to law school. For me, it has helped to read news stories and good forums like HN. For example, I know that non-compete clauses in employment contracts are unenforceable in California, because so many people have said so.
There are some lawyers who post here--grellas for example--and AMAs from lawyers on Reddit are helpful too.