Learn Farsi: سال گذشته، بودجه‌ی سرانه‌ی مهاجرها نسبت به قبل کمتر شد. - Last year, the per capita budget of migrants became less compared to before.

nesbat be means "relative to" or "compared to." It is a more formal and precise way to make comparisons in analytical, news, or academic texts. You could say az ghabl (than before), which is grammatically correct and very common in spoken Farsi, but nesbat be ghabl fits the formal, economic tone of this sentence much better.

What is the exact structure of the phrase budjeh-ye saraneh-ye mohajer-ha?

This is an ezafe chain linking three components. budjeh means "budget." saraneh is an adjective meaning "per capita" or "per head" (derived from sar, meaning "head"). Finally, mohajer-ha means "migrants." It literally strings together as "budget [of] per-capita [of] migrants." Notice that because budjeh and saraneh both end in a silent 'h' (ه), they must take the connecting sound -ye (written here as ی) to link to the next word.

Why does the sentence use the comparative kamtar shod (became less) instead of just kam shod (decreased)?

While kam shod (decreased) is a completely valid compound verb, when you explicitly set up a comparison using a phrase like nesbat be ghabl (compared to before), it is more natural to use the comparative adjective kamtar (less). It mirrors English: rather than saying "compared to before, it became little," you say "compared to before, it became less."