Learn Italian: È il pesce che è freschissimo. - It is the fish that is very fresh.

This is a cleft sentence (frase scissa). We use the È... che... structure to put heavy contrastive emphasis on the fish. It implies "It's the fish that is very fresh (as opposed to the meat or the vegetables)". If you just said Il pesce è freschissimo, it is a simple statement of fact without that strong, exclusive emphasis.

Since il pesce is the subject, could I use an infinitive construction here instead of che?

Yes! In cleft sentences, when the emphasized element is the subject of the verb that follows, you can use the implicit form with a + infinitive. So, È il pesce ad essere freschissimo is a perfectly natural and common alternative to this sentence.

Why is there an "h" in freschissimo?

The adjective fresco ends with a hard "c" sound (like a "k"). When you add the absolute superlative suffix -issimo, the "c" would normally soften before the "i" (making a "ch" sound, like in "cheese"). To preserve the original hard "k" sound of the root word, Italian spelling requires you to insert an "h".

How would I say "It was the fish that was very fresh" in the past tense?