Learn Russian: Холодильник пустой, нам совершенно нечего поесть. - The fridge is empty, we have absolutely nothing to eat.

Because нечего is a predicative word that describes a state of having nothing to do or consume. In Russian, this state happens to someone. So, the logical subject "we" takes the Dative case (нам), followed by нечего and the infinitive поесть. It literally translates to "To us, there is nothing to eat."

What is the difference between нечего in this sentence and the standard word for nothing, ничего?

Нечего is used specifically with an infinitive to say that there is nothing available to do. It always takes the stress on the first syllable (NE-che-vo). The standard word ничего (ni-che-VO) is used in regular negative sentences, like мы ничего не едим (we are eating nothing), which requires a conjugated verb and an extra не.

Why does the sentence use the perfective verb поесть instead of the imperfective есть?

In constructions with нечего + infinitive, we use the perfective aspect when talking about a specific, one-time action. The speakers want to eat a meal right now, but cannot. If you used the imperfective есть, it would sound more philosophical or refer to a general, ongoing lack of a food supply.

What does совершенно mean here, and how does it fit into the sentence?

Совершенно means "absolutely" or "completely." It pairs naturally with нечего to emphasize that the lack of food is total. It highlights the direct consequence of the first clause: the fridge is totally empty, so there is absolutely nothing to eat.