How is language symbolic?

Depends on what you mean by symbolic. However, language can be symbolic in several ways:
1) Language is a symbol or testament to one's culture. People think differently in other languages. People have different beliefs. Not everything translates the same. There are words in some languages that do not correspond with any word in another language. Translations really are difficult. Thus, language could be a symbol for culture.

2) Language has symbolism. This is common in literature. A rose is often a symbol for passion in English. Symbols are diverse.

3) Language is symbolic of intellect, from my interpretation. Ever notice that humans are much further ahead than any species? That's because our ability to communicate through language is much more powerful than most creatures. We have abilities that accompany this, but our ability to communicate makes us very special. Suppose you could not convey anything to the rest of the world. You would be alone. There would be no globalization. Another special thing we have about language is the fact that we have so many different ways to communicate. We communicate through math, music, movement, writing, speaking. We communicate in so many different ways. Thus, it is a symbol of our intellect, and if you dare say-superiority.

4) All language is metaphor. It is a sound or a set of graphic symbols that represents something else. Words don't really take part in the existence of the things referred to. Think of the word Apple, or the word information. What are they? How many meanings could they have? We could as easily call them phlimbara or quinch.