Learn Russian: Его улыбка очень приятная.

Unlike possessive words like мой (my) or наш (our), the third-person possessives for 'his' (его), 'her' (её), and 'their' (их) never change their endings. They stay exactly the same no matter the gender, number, or case of the noun they describe.

Where is the verb 'is' in this sentence?

In Russian, the present tense of the verb 'to be' is almost always skipped. You literally just say 'His smile very pleasant', and the 'is' is naturally understood from the context.

Why does приятная end in -ая?

It is an adjective describing улыбка (smile), which is a feminine noun. Because улыбка is the subject of the sentence (in the nominative case), the adjective must be in the feminine nominative form, which typically ends in -ая.

The translation given is 'His smile is pleasant', but doesn't очень mean 'very'?

Yes, очень means 'very', so a more exact translation is 'His smile is very pleasant'. Sometimes provided translations are slightly adapted, but the Russian sentence definitely includes the extra emphasis!