Learn Italian: Le sue parole hanno avuto un grande effetto.

You would add così or talmente before the adjective, followed by da and an infinitive verb. Notice that the word order usually shifts to put the noun first so the adjective and da sit next to each other: Le sue parole hanno avuto un effetto così grande da farmi piangere.

The translation says "His words", but couldn't Le sue parole also mean "Her words"?

Yes! In Italian, possessive adjectives agree in gender and number with the thing being possessed, not the owner. Because parole is feminine plural, the possessive must be sue, regardless of whether the speaker is a man or a woman. You would only know it means "his" here from the wider context.

Why is it un grande effetto and not un grand'effetto? Doesn't grande drop its final letter before a vowel?

Using an apostrophe before singular words starting with a vowel (like un grand'effetto) is very common, but keeping the full word grande is also perfectly correct. You only strictly have to shorten it to gran (without an apostrophe) before masculine singular nouns starting with a consonant, like un gran problema.

Could I use fare instead of avere here, as in hanno fatto un grande effetto?