The influence you can trade on the diplomacy screen has nothing to do with this (well, almost nothing - close enough to ignore).
What you need is to get your relations to Close. Do this by checking your information on the target civ; there is one screen in the foreign relations tab that gives you a breakdown of how that civ sees you and gives you a list of + and - factors that are affecting this. Check the list, and try to eliminate as many of the - factors as possible. To gain + factors, you can send freighters their way, give them econ or research treaties, build up your military, have the same ethical alignment, etc. Giving gifts can also help (that's the one way influence points can be used to affect this).
All of the preceding can only get your relations up to Friendly. The jump to Close has a random chance of occurring each turn you hold your relations at Friendly, so just keep checking every few turns. Nothing you can do will affect that jump.
Once you get the Close relation, BOTH civs need to have the Alliance tech to form an alliance. If they don't have it, sell it to them.
What does it mean when I actually give influence to the alien? What does it mean when the alien too has a slider for influence for deciding the amount to be given? What are they used for?
The influence points in the trade screen are the accumulated points your (and their) civ have generated since the last UP meeting. When you give them influence points, you are trading away a tiny bit of your voting power in the next UP vote. It's nearly impossible to trade enough of these points to affect the outcome of a UP vote, but they can be sold for cash in a pinch. Just remember that the AIs pay less for points when you offer more points at a time. You will get more cash selling 100 at a time than 1000, and 10,000 will get you very little. What the AI is willing to pay is also related to how much cash they have; getting their last 100bc is expensive and hardly worth your time.