I'm not sure what exactly in plain English means, but here is a try:
Think about it as consuming water.
Look at condition 1. If you have no water, you would die, making you extremely unhappy.
If someone came up to you and offered one bottle of water, it would make you infinitely happier than having no water, as you would live! (You would pay any amount of money to get that single bottle!)
The second condition states that eventually giving you extra water gives you less and less extra happiness. That first bottle was life saving, the second helps a lot,... but after the thousandth bottle, having extra water is nice, but having a thousand bottles versus a thousand and one does not increase your happiness too much.
The implication of the assumptions ensure that if I asked a consumer how much water she would like (for some price usually), she would pick a number between 0 and $\infty$.
She wouldnt pick 0, as we argued before, and eventually she cares less and less, so she would not buy $\infty$ bottles.
A famous example is $U(c) = \log(c)$.