I tend to understand "the street" to mean any defense situation that doesn't occur in the dojo, and is therefore likely to involve someone who may really want to hurt you.
(As oppossed to a partner doing a technique knowing full well it's my go next...) 'the street' isn't necessarily a physical street, it could be a bar, a house, etc, etc.
Why is it important? Well for us, a lot of what we practise is to do with the long term aim of making our aikido better, softer, more controlled and controlling. it isn't about learning to snap people.
However in a genuine self defense situation - one that we haven't been able to avoid by not going to the wrong pub, in the wrong part of the wrong city - the rules of the encounter are different.
for e.g - If someone come at me with a knife and i apply an ikkyo style move and they turn their back on me, the way a lot of people do when the move is first applied, I am going to use the opportunity presented to me (exposed ribs and kidneys etc) in a way i would never actually do in a dojo (although i might point it out when training with beginners who haven't learnt to uke.) I may also choose to kick down on the side an exposed knee, or a whole range of other things that wouldn't be done in the dojo becuase they are not safe for the attacker.