What's more awesome than neutron stars and red giants? Neutron stars inside red giants!
A Thorne-Zytkow Object is the end result of a merger between a Neutron Star and an O-Star (Red Giant). The two stars begin as a binary star system, and then the Neutron star (>1.4 solar masses) falls into the Red Giant (10-15 solar masses). This may seem strange that a star could fall into another star, but when we exam the densities, it doesn't seem so strange at all. The Neutron star is extremely dense because it is made exclusively of Neutrons. It has a similar demsity of an atomic nucleus. A single tablespoon of it is comparable to a fraight train filled with bricks. The Red Giant on the otherhand is very large, and so its mass is distributed over a greater volume, making for a much less dense object. A Red Giant has a density comparable to that of water. If it is easier, imagine the merger between a Neutron star and a Red Giant as being a rock sinking in water. In this process however it takes 1,000 years to get to the core and one month to get inside the core. The motion of the neutron star within the envelope of the red giant is on the order of Mach 3-1.4. TZOs are usually understood as a neutron star within the core of a red supergiant with the radii of several AU, a low temerpature, and a luminosity of order 10^5 solar luminosity units.
It is believed that the TZOs are formed at at a rate of 1/500 years to 1/1000 years per 10^11 solar mass Milky-way-like galaxy. This means that more than 30 TZOs could be formed per year within a 30 Mpc radius.
There are generally two possible outcomes of a neutron star sinking into the depths of a red giant. It will either become a supernova or a TZO.
That's a heavy meal.