Goti langu linaanza kuvimba baada ya kuanguka.

Because possessives have to agree with the noun class of the noun they describe.

• goti = knee
• magoti = knees

goti is in noun class 5 in the singular, and class 5 uses la- before the possessive stem -angu.

So:

• goti langu = my knee
• jicho langu = my eye

By contrast, yangu is used with other noun classes, for example:

• ndugu yangu = my sibling / my relative

So goti yangu would sound wrong.

linaanza can be broken into:

• li- = subject prefix for a class 5 singular noun
• -na- = present tense
• -anza = begin / start

So linaanza literally means something like it is starting or it starts, where it refers to goti.

Because in Swahili, the verb already contains that information.

In English, you need separate pieces:

• it
• is
• starting

In Swahili, those ideas are packed into one word:

• li-na-anza

So linaanza already means it is starting.

After verbs like kuanza (to begin/start), the next verb is usually kept in the infinitive form.

So:

• kuanza kuvimba = to start swelling
• kuanza kulia = to start crying
• kuanza kutembea = to start walking

Here:

• linaanza kuvimba = it is starting to swell

This is very similar to English start to + verb.

It can cover both ideas depending on context.

• kuvimba can mean to swell
• in some contexts it can also describe the state to be swollen

In this sentence, because it follows linaanza (is starting), the natural meaning is:

• is starting to swell

So the focus is on the process beginning.

baada ya means after, and it is commonly followed by:

• a noun, or
• an infinitive verb

Here it is followed by the infinitive kuanguka:

• baada ya kuanguka = after falling / after a fall

This is a very common Swahili pattern. The subject of falling is often left understood from context.

Yes. This is an important distinction.

• kuanguka = to fall
• kuangusha = to drop something / cause something to fall

So:

• Nilianguka. = I fell.
• Niliangusha simu. = I dropped the phone.

In your sentence, kuanguka is correct because the person fell; they did not drop the knee.

Yes. That would still be natural.

• Goti langu linaanza kuvimba baada ya kuanguka.
• Baada ya kuanguka, goti langu linaanza kuvimba.

Both are fine. The second version puts more attention on after falling at the start, similar to English.

The difference is mainly one of aspect:

• linaanza kuvimba = it is starting to swell / it starts swelling
• limeanza kuvimba = it has started swelling

So:

• li-na-anza uses the present
• li-me-anza uses the perfect

If you want to emphasize that the swelling has already begun, limeanza kuvimba is often a good choice.

The plural of goti is magoti, and the agreement changes because magoti is class 6.

So:

• Magoti yangu yanaanza kuvimba baada ya kuanguka.
  = My knees are starting to swell after falling.

Notice the changes:

• goti → magoti
• langu → yangu
• linaanza → yanaanza

That is a good example of how noun class agreement affects both possessives and verbs.