Thermal paste is, as you probably already know from the video, is a glue-like substance that conducts heat fantastically well. It is used to dissipate heat from processors into heat sinks as well as other heat generating objects to their heat sinks.
What the person in the video did (at least what I got from quickly scrolling through and your description) is he added some metal to the phone and used thermal paste between it and the phone's processor. This will lower the temperature of the processor and make it run a little bit better.
In my opinion, in a phone, this isn't a good idea. In PCs where there is a dedicated heatsink already with a cooling system on it, there is room for this to be effective and for the heat sink to dissipate the heat. In a phone, the heat will not have any more room to dissipate than it previously did. There is no method of cooling the heat sink. I'd actually be concerned that areas of the phone that were not designed to experience such heat could become problematic when the makeshift heat sink becomes hot, such as the plastic coated ribbon cables that likely have low melting temperatures. In addition, phones are not designed to be tampered with like this. You are far more likely to damage the phone than help it.
That's my 2 cents.