Learn Spanish (Spain): A ellos les gusta el baloncesto.

In Spanish, gustar translates closer to "to be pleasing to". The verb must match the thing that is pleasing, not the people who like it. Since el baloncesto (basketball) is a singular thing, we use the singular verb gusta. You would only use gustan if they liked a plural thing, like "the shoes" (los zapatos).

Why do we need both A ellos and les in this sentence?

The pronoun les (to them) is required when using gustar. The extra phrase A ellos is added to clarify exactly who les refers to, since les could also mean "to you all" (ustedes). It can also be used just to add emphasis.

Can I leave out A ellos and just say "Les gusta el baloncesto"?

Yes! If it is already clear from the conversation that you are talking about "them", you can completely drop A ellos. However, you can never drop the pronoun les.

Why is there an el before baloncesto? We don't say "the basketball" in English.

When talking about general categories, hobbies, or sports you like, Spanish requires the definite article (el, la, los, or las). Because of how gustar works, you are literally saying "the basketball is pleasing to them."