What you are looking for is three separate features, which cannot be clubbed into one app or software.
For taking a Screenshot Mac has an inbuilt capability and function to take screenshots as mentioned by Damilola below.
For blurring an image you can use a blur tool available in a Photo and Image editor like – PhotoPad
Its as simple as downloading PhotoPad, opening the image and selecting the Blur option under the ‘Effects’ Tab.
You can then select the type of blur you want and the intensity of Blur.
The below sample is for a Zoom Blur at 42 intensity.
For a Screen recorder – you could try Debut – Screen Recorder and Video Capture
Debut lets you record and capture video from almost any source – external device, VHS, webcam or your computer screen.