adverbs after linking verbs

In your first sentence the copular verb (or linking verb) is followed by an adjective complement. It links the subject 'food' to its adjective complement 'sweet'.

The food tastes sweet.

Other examples might be

She is happy. (not happily)
The laboratory smelled awful. (not awfully)
He seems nice (not nicely) etc.

So the point your grammar book is making is that the copula is followed by an adjective complement, unlike other verbs which are followed by adverbs (e.g. The dog ran happily.)

In a further step you could modify the adjective 'salty' with an adverb and write:

The food tastes intensely salty.

Here 'salty', the adjective, is modified by the adverb 'intensely'. If you change the adjective 'salt' into a noun, then you need to say:

The food tastes intensely of salt.

This is the same structure as your last sentence:

The laboratory smelled strongly of chemicals.

In English the primary copular verb is be. The function of a copula is to link the subject of the sentence with its complements. The subject will always be a noun or noun equivalent and its complement an adjective. There are certain other verbs such as become, get, feel, look, taste, smell, and seem that can have this function.

We can always replace these copular verbs with the primary copula be, if they are functioning as a copula and this process can also be used as a test method to ascertain whether an adjective or adverb should be used.

E.g. The food tastes sweet. (The food is sweet.)

The laboratory smells strongly of the chemicals. (We cannot say that *'the laboratory is strongly. So, the verb smells is not a copula here. It is an action verb.)

He became a student. (He was a student)

They look tired. (They are tired)

The milk smells good. (The milk is good)

(This usage should be distinguished from the use of some of these verbs as "action" verbs, as in They look at the wall, in which look denotes an action and cannot be replaced by the basic copula are.)

Wikipedia.

"to smell" as "to have a particular smell" can be only a linking verb.

The verb smell can take a prepositional phrase headed by of as its complement in order to express that something has a certain smell. For example, "The laboratory smelled of chemicals" is a valid sentence that expresses the idea that the laboratory had a particular smell: the smell of chemicals.

In "the laboratory smelled strongly of chemicals", the adverb "strongly" simply serves to emphasize the extent to which the laboratory smelled of chemicals.

(Neither "The laboratory smelled strongly" nor "The laboratory smelled strong" make much sense to me. I guess you could say "The laboratory had a strong smell." It's true that we say "The laboratory smelled bad" and not "The laboratory smelled badly".)