How is it possible for RNN to do sentiment analysis?

It depends on how you configure your model and the layers you're using. You can add other layer's of DNN and experiment with those.

You should also consider using LSTM(Long short-term memory), since the past information do not get decayed quickly like in RNN, and you keep the present values, thus having a better model based on RNN.

You can also you a Bidirectional LSTM (BLSTM), which improve the learning process by using a cell state that runs through all your past data and generalize well for sentiment analysis.

Keep in mind that RNN cannot detect negation in sentences, like saying "This movie was not good", usually ANN need more configuration to understand those kind of aspects.

Another trick is to add a CNN layer to your model, so it can extract key words, thus when your data is forwarded to the RNN based layers, it will less noisy and more oriented to specific keywords.

RNNs do not learn to predict sentiment. They learn correlations between the input data and the target labels. If they see that every time the input contains the word "bad" they have to generate the label "negative", then they will learn it. If they see in the training data that the previous phenomenon happens always except when there is a "not" before "bad", then they will learn it. Depending on how the data distribution is, they may generalize the negation to any combination of verbs and adjectives, or maybe not, and they only handle negation appropriately when the input data is very very similar to the training data.

Your question seems to aim at understanding the "internal dynamics" of the RNN when it is predicting sentiment. While there is some research in that direction, I think they can still be considered as "black boxes", in that we do not actually understand the functions modeled by neural networks and therefore their outputs are not "explainable"

Your problem is a sequence classification exercise. Yes, RNNs are very much needed to summarize the sentence intent as a whole. The last state of the RNN is fed to a Dense layer to predict the sentiment. This works nicely. You can further improve the exercise by adding 'Attention' by using all the hidden states of the RNN while predicting the output and paying attention to the ones that really matter.

See for e.g.: https://stackoverflow.com/questions/63060083/create-an-lstm-layer-with-attention-in-keras-for-multi-label-text-classification/64853996#64853996