Learn Hardcore Swahili: Msichana anapenda kucheka.

Msichana = “girl”
a- (subject prefix) = “she” (3rd person singular)
-na- (tense-marker) = present tense (“is …ing” / “likes”)
-penda (verb root) = “like”
ku- (infinitive marker) = “to”
-cheka (verb root) = “laugh”
So a literal gloss is “girl she-present like to-laugh,” i.e. “The girl likes to laugh.”

Why do we use kucheka instead of just cheka here?

In Swahili, when one verb is the object of another (like penda “like”), the second verb must be in the infinitive form. The infinitive is formed by adding ku- to the verb root.
• cheka = “laugh” (root form)
• kucheka = “to laugh” (infinitive)

How does anapenda show that the action is happening in the present?

Swahili marks present tense with the infix -na-, placed between the subject prefix and the verb root:
• a- (she) + -na- (present) + penda (like) = anapenda (“she likes” / “she is liking”).

Do I have to include Msichana if the verb already has the prefix a- for “she”?

No, the independent subject noun is optional and used for clarity or emphasis. You can simply say:
• Anapenda kucheka. = “She likes to laugh.”
But if you want to specify “the girl” (rather than “he” or “she” generically), you keep Msichana.

How would you say He likes to laugh in Swahili?

Swap the subject prefix:
• Anapenda kucheka.
Here a- on anapenda can mean “he” or “she.” For emphasis you could add yeye:
• Yeye anapenda kucheka.

How do I change the sentence into the past tense (“The girl liked to laugh”)?

Replace the present tense marker -na- with the past tense marker -li-:
• Msichana alipenda kucheka.
(Here a- = she, -li- = past, penda = like.)

Where do I place sana if I want to say The girl likes to laugh a lot?

Adverbs like sana (“very / a lot”) usually follow the verb or the infinitive. Two common options:
1) Msichana anapenda kucheka sana.
2) Msichana anapenda sana kucheka.
Both mean “The girl really/a lot likes to laugh,” with a slight nuance:

Version 1 emphasizes frequent laughing.
Version 2 emphasizes how much she enjoys it.

Is kucheka acting as a noun in this sentence?

Not exactly as a noun, but it is a non-finite verb (infinitive) functioning like a noun phrase (the object of penda). In English we say “like to laugh”; in Swahili you simply use the infinitive kucheka.