How does a Dual-Antenna WiFi router work better in terms of signal strength?
In free space with a nice direct path, it would make little difference.
Unfortunately, WiFi doesn't operate in free space: it operates in a complicated sea of reflections from walls, people, furniture, wiring etc. The reflections add together in more or less random phases, and from time to time the signal level can drop quite significantly. This is called Rayleigh Fading.
As the fade depends on the relative phases of the signal paths, being only half a wavelength away is enough to make the random fades at these two positions completely independent. If there is a 10% chance of each antenna having a signal strength too low for operation, then the chance of both antennas suffering is just 1%. This is called Diversity Reception. The receiver decodes both channels, and uses the one with the fewest errors.
There are other, cleverer, more complicated ways to use two antennas in reception and transmission to improve the signal strength or data rate, but this one is the easiest to understand.
With two or more antennas it is possible to use MIMO (Multiple Input, Multiple Output) but also to use Diversity, see this article.
I do not think MIMO by itself enhances the range or signal strength. However Diversity does as it uses the two antennas in such a way that the signal is aimed in a certain direction.
This works by sending a slightly different signal to each antenna and through phase cancellation and adding, at some locations a stronger signal can be received.
It works in the other way as well, with two receiving antennas the phase differences between the signals are used to cancel out or suppress unwanted signals.
Do note that you're comparing a one antenna router versus a two antenna router. It is very well possible that the two antenna router is also more modern and is simply more sensitive than the older router. It cannot send a stronger signal though, there are regulations which dictate how much signal can be send by a router, no matter how many antennas it has.
Regarding noise cancellation: as noise is random and cannot be cancelled it is better to speak about interfering and disturbing signals. Lowering the levels of these signals does help as that increases the signal-to-disturbance ratio (and thus the signal-to-noise ratio, if we call anything we do not want "noise") of the signal you want to receive. A higher signal to noise ratio allows for a more advanced modulation scheme to be used (for example QAM 64 instead of QAM 8) so the bitrate can be higher.