Matangazo tuliyoyasikia kwenye redio yalikuwa muhimu.

Matangazo is the plural form of tangazo.
So:

• tangazo = announcement / advertisement / notice
• matangazo = announcements / advertisements / notices

It is a noun class 6 plural, and its singular is noun class 5.

Swahili does not have articles like a, an, or the.

So matangazo can mean:

• announcements
• the announcements
• sometimes even some announcements

The exact sense usually comes from context.

Tuliyoyasikia can be broken down like this:

• tu- = we
• -li- = past tense
• -yo- = relative marker agreeing with matangazo (class 6), meaning something like which/that
• -ya- = object marker agreeing with matangazo (class 6), meaning them
• -sikia = hear

So literally it is something like:

we + past + which + them + hear

A natural English rendering is simply that we heard.

They do two different jobs.

• -yo- connects the verb to the noun matangazo and makes it a relative clause: which/that
• -ya- is the object marker inside the verb, referring back to matangazo: them

So Swahili is being more explicit than English.
A very literal version would be:

the announcements which we heard them

English does not say that, but Swahili commonly does in this structure.

Because in this sentence, Swahili builds the relative meaning directly into the verb.

Instead of a separate word, the verb carries the relative marker:

• tuliyoyasikia = that/which we heard

This is very common in Swahili.
So the idea of which is inside the verb form itself.

Yes. You could also say:

Matangazo ambayo tuliyasikia kwenye redio yalikuwa muhimu.

That means the same thing.

Here is the difference:

• tuliyoyasikia = compact built-in relative form
• ambayo tuliyasikia = relative meaning expressed with a separate relative word

Both are correct. The version in your sentence is a little tighter and very natural.

Because matangazo is a class 6 noun, and the verb has to agree with it.

Yalikuwa breaks down as:

• ya- = subject marker for class 6
• -li- = past tense
• kuwa = be

So yalikuwa means they were.

The ya- is there because the subject is matangazo.

Because muhimu is one of the descriptive words that usually stays the same across noun classes.

So you get:

• tangazo muhimu = an important announcement
• matangazo muhimu = important announcements

Unlike some adjectives in Swahili, muhimu does not need a class prefix here.

Kwenye redio means on the radio.

Kwenye is a locative word that can mean:

• in
• on
• at

The best English translation depends on context.
With redio, the natural English choice is on the radio.

Redio is a loanword that has been adapted to Swahili spelling and pronunciation.

Swahili often writes borrowed words in a way that fits Swahili sound patterns, so redio is the normal form in Swahili.

It is usually treated as a noun whose form does not change between singular and plural.

It would be:

Tangazo tulilolisikia kwenye redio lilikuwa muhimu.

That means The announcement we heard on the radio was important.

Notice how the agreement changes:

• tangazo instead of matangazo
• tulilolisikia instead of tuliyoyasikia
• lilikuwa instead of yalikuwa

That is because the noun has changed from class 6 plural to class 5 singular.