You normally have to do this at the client end, e.g. in your code, by parsing the HTML and then using the W3C DOM method getTextContent on the root node (for example).
However in reviewing the documentation on Rich Text fields, it seems that if you reference the field in a Formula the HTML markup is stripped out.
When a rich text area field is used in a formula, the HTML tags are stripped out before the formula is run. For example, when a rich text area field is used in a validation rule’s criteria, the HTML tags are removed before the evaluation.
You cannot directly reference a Rich Text field in a regular Formula Field, however you can reference it, in addition to validation rules, in a Workflow Field Update. In the example below I created a "Rich Text" field (Rich Text type) and "Rich Text Text" field (regular Text Area type). Then I create a Workflow and Workflow Field Update to update the Rich Text Text field each time the user changed the Rich Text field.
With something like this in place you can query for the text only version in your REST request. The following shows my Workflow definition if you need to understand this further.