There are a number of factors that might cause this kind of issue:
Your website's cache
The concept of caching consists in storing an already-computed webpage on the server and passing it on to the clients without having to generate it again every time it gets requested. Although this method greatly increases the overall performance in many situations, it can sometimes create problems as well, especially when it comes to forms.
Please refer to the My forms are not working correctly with cache enabled article for more information on how to fix this.
Missing form component
If you've edited the form's layout you should know that, in doing so, you first had to disable its automatic generation. Make sure that you haven't removed any form component that might have been set as required (Components >> RSForm!Pro >> Manage Forms >> edit your form >> Components >> edit the field >> Validations >> set Required to Yes). If this is the case, either re-add the field's placeholders in the form's layout, or set the field to be not required.
Hidden validation error
One (or more) of your fields may have not passed validation (if you've set them up so that they have a validation enabled) and, for some reason (CSS code that's been overwritten, you have removed the field's validation placeholder from the form's layout) the validation message does not get displayed.
Script in Email field
If you notice that, after clicking on Submit, the form doesn't get submitted and the email field gets filled with a script, know that this is being generated by a malfunction of one of Joomla!'s security features, email cloaking.
Please refer to the My email field contains a script article for detailed explanations on how to correct the issue.