What does 憋得通红 mean here?
*憋 means 'to hold back'--  most commonly: 'holding breath' (憋氣) or 'holding back urine' (憋尿)

In the case of '脸憋得通红', '憋' should be short for 憋氣 (holding breath), (too upset to breath)

脸(face) 

憋(holding back)

得(to the point of)

通(completely)

红(red)

To translate this phrase literally would be  'face turns red from holding back' . 

If you want to be precise, you can write 'face turns red from holding breath'. but it is not good English.

I think the best way to translate this phrase is to interpret the reason of holding breath directly, and write: "face turns red from embarrassment" (having difficulty to breath due to embarrassment)

Similarly, if the reason for the redden face was 'anger' the translation should be "face turns red from anger"