When playing I notice that within a relatively few amount of turns, 10-15 turns, the computer automatically leaps to the top of the scoreboard. Why is that? I understand they are AI and they are going to have advantages but what specifically does the AI do that players don't? I observed on the scoreboard that the AI grows out there cities abnormally quick i.e. second city gets built fast, population grows faster, land gets acquired faster. Do I have to develop military to beat them?
AI "cheats" on the highest difficulties by getting boosts to stats, such as Science. If you are playing on a lower difficulty, they might simply focus more on food to make their cities grow faster, or they might be doing actions which give higher scores, but are not necessarily more beneficial. Score is only a general indication, don't worry about it.
How to display score breakdown:
If you go to the Diplomacy screen (where the scores are shown), you can hover with your mouse over the score of one player. When you do that, the game will display a breakdown of what factors are causing the score to be what it is.
Reason for computer's high scores:
And as Rokk says, the major reason for the effect you are mentioning is probably AI bonuses based on difficulty level. I tried playing to turn 15 in one game on Apollo difficulty (big bonuses for AI), and one game on Mercury difficulty (no bonuses for AI). These are the breakdowns of scores of the best scoring computer opponents at turn 15 (out of the ones that had landed at turn 15).
Apollo opponent score at turn 15:
Total : 53 From cities : 4 From population : 7 From land : 6 From technology : 24 From virtues : 12
Mercury opponent score at turn 15:
Total : 31 From cities : 4 From population : 7 From land : 4 From technology : 8 From virtues : 8
The most notable difference is that the apollo level AI probably got to start with a couple of free technologies in the beginning, judging by his high technology score.
Notice also, that my score varied a bit between the two games also. In one, I had 33 points and the other 27 points. So some score variations might just be random differences. That may depend on many different factors. The terrain around your city might influence the growth rate, causing different points from population. You might be lucky/unlucky with expeditions/resource pods, causing you to gain technologies/virtues more quickly/slowly etc.