Inflluence points are used for voting in the UP, not for flipping planets.

To influence flip planets, you need to project cultural influence onto them. The simplest way is to use your own planets - large populations and influence boosting projects will project more influence. But this isnt likely to flip enemy planets on its own, especialy not in other starsystems. For that, you need the influence starbase, and you need the higher tech influence modules as well. Build these near enemy planets, and build enough that the planet is subject to four times more of your influence than its own civilisation. After that, it'll be marked with an icon denoting a chance of flipping, and eventualy it will flip. How quickly depends on exactly how much influence you project, but its a random roll each turn. Sometimes youll get it on the first or second turn, sometimes they dont flip for hundreds of turns.

Also, try and grab an influence resource or two.

Be warned, the AI will spot you doing this and may or may not take action against you. A high military power will help you keep them peaceful.

Flipping planets tends to be an all or nothing strategy-- either you have a lot of influence and the planets fall like dominoes, or they don't fall at all.

To win with "soft power" try the following:

Make sure you have large populations on all the worlds that can support them. Influence is predominantly generated by population.
Get the restaurant of eternity development.
Research all the techs that give influence bonuses (mostly in the trade and goverment tech trees).
Get any influence resources and mine them all out.
Build influence mines-- as mentioned above this may provoke a violent reaction if you're diplomacy & military ratings aren't high enough.

Adding to the above. I didn't build a single farm either. I kept moral at 75 until I could afford to get it up to 100. Once it was at 100 for a few turns, the cash increased. Other than the minor races, I have not been in another war. And the minor race was taken out in one turn (to make sure ALL my tech didn't get away from them).

Here is what I build:
Factory
Factory
Moral
Research only on bonus
Econ Centers everywhere else.

If the planet is small I only build 1 factory, then overwrite it later with tech or money.

I only build starports on a few planets, because everyone elses planets will have a starbase, and when they flip, they will be mine.

Hmm, I think I get it now. Do you need an influence resource to build an influence starbase or can you make one anywhere?

Like 1100 influence over this planet now, help meeee