Clueless about training hard and taking it easy

WONDER what truth there is to the claim that you should rest enough after you've done some running? What is enough and how much rest should you take compared to how tired you are after exercising?

I have taken two days off after I couldn't run continuously for 10K with an average speed of 11 kph. I was just too tired from the training I had done earlier that week. I had to run for 9½ minutes and walk for 30 seconds. After four times I had to walk 10 minutes and continue at a cycle of 2 minutes running and 1 minute walking. My running speed was approximately 10.6 kph (according to my GPS watch).

Now what should have been better: train less intensive and keep training, or--like I did last week--keep training intensively and take two days off to recover.

I don't have a clue.