Well, we can deal with one of these words very easily indeed. The 'odd one out' is 'steep'. 'Steep' is an adjective, used to describe the gradient of a slope. If a road or hill rises very quickly, and you have to go up it in 1st or 2nd gear, it is steep. If prices rise quickly over a short period, you can say that the price rises are steep. I've no idea why you've bundled this word in with the others!
Edge, brink and verge are more interesting. They are all nouns with a similar meaning, but different uses.
'Edge' is physical. It's mainly used for literal and concrete contexts : the edge of a table, the edge of forest, for example.
'Verge' and 'brink' are used in a more abstract way, often in the phrase 'on the brink of..' and 'on the verge of..', followed by a noun or a gerund. We are in the territory of collocation and idiom here, so there are no explanations. All you can do is learn some common phrases. For example, you could say that in the late summer of 1939, Europe was 'on the brink of war'. We can say that an economy is on the brink of collapse. We can also say 'on the brink of disaster' , 'on the brink of bankruptcy'. Getting the picture? It's generally about negative things which are just about to happen.
'On the verge' of is very similar. Probably the commonest collocation is 'on the verge of tears' when someone is about to cry. 'On the verge of ..' can be used in the same way as 'on the brink of', but it's less dramatic. We can often use 'verge' for more everyday situations than 'brink'. For example, you can say 'I was on the verge of accepting the offer, but then I changed my mind'. By the way, a few hundred years ago 'verge' had the same vulgar meaning as it does in French, but it doesn't any more. Honestly.
I hope that all helps.