Why we need more intensions
Suppose we want a theory that tells us for all sentences in our language in what possible contexts their utterance is true. Call those functions from contexts to truth values "A-intensions". A systematic theory should tell us how the A-intension of complex sentences depend on their constituents. Here are some theories which are not very satisfactory in this respect.
Theory 1. Each sentence consists of a sentence radical and a fullstop. (The sentence-radical is the entire sentence without the fullstop.) All sentence radicals have the same semantic value: God. The semantic value of the fullstop maps this semantic value to a truth-value. But whether it maps God to true or false depends on the context of utterance. For instance, in a context in which it doesn't rain and the utterance of "." is preceeded by an utterance of "it rains", the value of "." maps God to false; in a context where "." is preceeded by "2+2=4", it maps God to true; and so on.
Theory 2. We assign God as semantic value to all singular terms and, for any context, some set of things to all predicates. A sentence "A is F" is true in a context C iff God (the value of "A") is a member of the set assigned to "F" (and all other predicates). Which set that is depends on the context C. For instance, in a context where "Paris is in France" is uttered, the value of "is in France" (and any other predicate) is the unit set of God; in a context where "Berlin is in France" is uttered, it is the empty set. If one utters "Paris is in France and Berlin is in France", the context switches mid-sentence. Don't ask about "Paris and Berlin are both in France", and don't ask about intensional operators.
Theory 3. To every singular term we assign some mereologically unextended object, and to every predicate some set of such atoms. "A is F" is true in a given context iff the value of "A" is a member of the set assigned to "F". Again, which set that is depends on the context. Suppose, for example, that "Berlin" and "Ralph" are assigned the same atom A in Ralph's left ear. Then in a context where "Ralph is a rabbit" is uttered, the value of "is a rabbit" contains A, but not in a context where "Berlin is a rabbit" is uttered. If one utters "Ralph is rabbit and Berlin is not a rabbit", the context switches mid-sentence. Don't ask about "Ralph and Berlin are both rabbits". For intensional operators, see theory 4a or 4b.
Theory 4a. Assume we have somehow assigned truth-values to every atomic sentence in every context, but our language also has sentential operators like "somewhere", "sometimes", "possibly". In any context, the semantic value of "possibly" turns a semantic value of the embedded sentence into a truth value. So let's say that the value of "possibly" maps truth-values to truth-values. This mapping, the semantic value of "possibly", is context-dependent: in a context where it doesn't rain and the embedded sentence is "it rains", the value of "possibly" maps false to true; in a context where the embedded sentence is "2+2=5", it maps false to false; and so on. (Lewis ignores this contextualist solution in his discussion of the 'extreme external strategy' in Plurality, pp.43f.)
Theory 4b. Suppose we've assigned truth-values to atomic sentences in contexts by assigning individuals to singular terms and corresponding sets to predicates. Then we can let the value of "somewhere", "sometimes", "possibly" operate not on the net semantic value of the embedded sentences, but rather on the semantic values of its constituents (a strategy Lewis always ignores in his semantic work). Let's say that "possibly, A is F" is true iff some counterpart of the value of "A" is in "F". Whether something is a counterpart of an individual depends on the context. For instance, suppose "the statue" and "the lump" denote the same individual A, and "is made of gold" a set S. Then in a context where "possibly, the statue is made of gold" is uttered, some member of S counts as a counterpart of A, but not in a context where "possibly, the lump is made of gold" is uttered. Similarly, if we endorse theory 3 and say that "Berlin" and "Ralph" are assigned the same atom A, we can say that "somewhere, Ralph is windy" is false but "somewhere, Berlin is windy" true because A has different spatial counterparts depending on which of these sentences is uttered.
What's wrong with these theories is that they ignore much of the systematicity in our language. Theory 1 is not compositional at all. It's just a complicated way of saying that in any context, any sentence has a truth value. We new that before. What we wanted was a systematic theory that allows us to compute this truth value given enough information about the context. Saying that a semantic value is context-dependent doesn't free us from the obligation to give a systematic account of this dependence (as long as there is one).
Consider theory 3. We can assume that the truth-value of "--- is a rabbit" depends systematically on what name of the atom A in Ralph's left ear fills the blank: if "Berlin" fills it, the sentence is false, if "Ralph" fills it, it is true. The same systematic dependence is observed for many other (corefering) names. But a systematic semantics should tell us which names generally combine with "--- is a rabbit" to form a true sentence and which do not. If reference does not distinguish these classes of names from one another, we need more than just reference in our semantic value. (We need something like C- and T- and P-intensions for singular terms, functions that tell us for different places at different times and worlds whether our atom 'exists at' that place and time and world. We also need those C-intensions to complete theory 4b, and corresponding C-intensions for sentences to complete theory 4a.)
It is sometimes said that context-dependent parts of meaning belong to pragmatics, not semantics. Then I believe the five theories above illustrate that semantics can be made arbitrarily simple by pushing all its work over to pragmatics. What matters is that the work is done.
(Perhaps some aspects of context-dependence cannot be accounted for in any simple, systematic way. The truth value of de re modal predications in particular seems to depend a lot on all kinds of background assumptions, the previous conversation, etc. Then we might leave that part of our theory of meaning blank, and say that the C-intension of terms depends in some unknown way on the context. Moreover, if the truth-value of de re modal predications does not vary much with the names employed as long as these names have the same T- and P-intension (in the relevant context), one could also drop the individual C-intensions of names entirely and go for something like theory 4b. But if one does the same for spatial and temporal predications, one leaves parts of the semantics blank that can easily be filled in.)