Reading. Critical thought. The kind of rudimentary stuff that literate and liberally educated adults find themselves capable of doing every day (if their curiosity drives them to it).
It's a fair question, asking why one "believes" in a scientific theory. On the other hand, it is rather inconvenient to have to answer this on a daily basis.
I'm sure that most people believe science simply because, in their daily lives, it appears to work. Quartz watches, computers, satellite TV, GPM, cell phones, medicine, dentistry. I personally see nothing very wrong with this. Even the most brilliant person cannot master all areas of knowledge. At some point we take things on trust.
I take evolution to be established in part because of trust, but also because I have spent decades reading about it, and because I see the principle of selection working in many ways, not all of which are biological.
Darwin says he got the idea for natural selection after reading the Scottish economists. That probably included Adam Smith. The invisible hand is a powerful metaphor. It is neither chance nor an intelligent agent. It is something not easily classified by traditional philosophy, an after the fact cause.
Selection neither pushes nor pulls. It constrains; it prunes. Because the conditions of selection are dynamic, the shape of the constraint cannot easily be quantified or described. It is a moving target. Just as no industrialist can guarantee a market for a new product, no imaginable designer can be certain what traits will be successful for the next generation of organisms. The reason I do not "believe" in intelligent design is I think it is impossible. Designing for all contingencies is as impossible as predicting the weather years in advance.