Reason for adding 1 to word index for sequence modeling
First, note that they are just adding 1 to the size of the vocabulary, not to the token IDs themselves, so the predictions are not affected.
Then, why adding 1 ?
Because Tokenizer.word_index is a python dictionary that contains token keys (string) and token ID values (integer), and where the first token ID is 1 (not zero) and where the token IDs are assigned incrementally. Therefore, the greatest token ID in word_index is len(word_index). Therefore, we need vocabulary of size len(word_index) + 1 to be able to index up to the greatest token ID.
Update: note that adding 1 to the vocabulary size has nothing to do with out of vocabulary words: the words that are not pretrained are encoded as the out-of-vocabulary token (oov_token) if it was provided when building the tokenizer, or ignored if not. The oov token, if provided, has index 1.