When a star becomes a black hole, does its gravitational field become stronger? I've seen in a documentary that when a star collapses and becomes a black hole, it starts to eat the planets around. 
But it has the same mass, so how does its gravitational field strength increase?
 A: If you measure the large-distance strength of the gravitational acceleration $g\approx \frac{GM}{r^2}$ of a star / black hole with the assumption that your distance $r$ is much further out than the various mass parts, shock wave, and ejected material; then $g\approx \frac{GM}{r^2}$ is (within a percent or so) the same before and after the supernova. This is so (i) because we can (to that precision) ignore the fraction of energy in the supernova coming from ultra-relativistic particles consisting mainly of neutrinos, and (ii) because of averaging effects similar to Newton's shell theorem and Birkhoff's theorem.
A: Actually, it doesn't have the same mass, it has significantly less mass than its precursor star. Something like 90% of the star is blown off in the supernova event (Type II) that causes the black holes.
The Schwarzschild radius is the radius at which, if an object's mass where compressed to a sphere of that size, the escape velocity at the surface would be the speed of light $c$; this is given by
$$
r_s=\frac{2Gm}{c^2}
$$
For a 3-solar mass black hole, this amounts to about 10 km. If we measure the gravitational acceleration from this point,
$$
g_{BH}=\frac{Gm_{BH}}{r_s^2}\simeq10^{13}\,{\rm m/s^2}
$$
and compare this to the acceleration due to the precursor 20 solar mass star with radius of $r_\star=5R_\odot\simeq7\times10^8$ m, we have
$$
g_{M_\star}=\frac{Gm_\star}{r_\star^2}\simeq10^3\,{\rm m/s^2}
$$
Note that this is the acceleration due to gravity at the surface of the object, and not at some distance away. If we measure the gravitational acceleration of the smaller black hole at the distance of the original star's radius, you'll find it is a lot smaller (by a factor of about 7).
A: When you watch a pop-sci TV show, you need to take everything you see with a very healthy grain of salt. This is particularly the case if the show's host isn't a scientist, but even when a scientist is the host, you need to be suspicious.
Stellar black holes do not turn into monsters that reach out and pluck objects from the heavens. From far away, a black hole behaves no differently gravitationally than an ordinary object of equal mass. It's only when an object gets very close that black holes behave differently. Note that this "very close" means what would be well into the interior of the ordinary object.
If anything, stellar black holes are little kitty cats rather than big monsters compared to the stars that generated them. The supernovae that generate stellar black holes blow away a very large portion their mass during the supernova event, both as energy and ejected matter. The resulting black hole has a much smaller mass than did the parent star.
If the parent star is a member of a close binary star, the black hole might still be able to draw mass from the other star. But reaching out and inhaling planets? That's just bad pop-sci.
Except for the outer atmosphere of a red giant that is a close binary pair of a stellar black hole, it would be quite amazing for a stellar black hole to gobble up anything. It would take a lot of energy to intentionally send something very close to a black hole. By way of analogy, mankind has sent four satellites out of the solar system (with a fifth on the way) but we have only sent two missions to Mercury. The reason is that takes a lot of energy (a whole lot of energy!) to get to Mercury. Escaping the solar system is a piece of cake compared to getting to Mercury. It would take even more energy to get very close to the surface of the Sun. If our Sun was instead a one solar mass black hole, it would take a whole, whole lot more energy to send something within a few Schwarzschild radii of the black hole.
A: It actually goes the other way around: when a star collapses to form a black hole, its planets (if it has any) will become unbound and fly away to infinity. 
Simple reason: when the star explodes to form a compact object (neutron star or black hole), it releases most of its mass in the form of a SuperNova explosion, so that the central object around which the planet is orbiting has a much smaller mass than the original star. The least decrease is roughly from an $8 M_\odot$ star to a $1.4 M_\odot$ neutron star, giving a minimum reduction of about a factor of 6. 
Now let us consider what happens to the planet. Before the explosion, its kinetic energy $K$ is half the modulus of its potential energy $W$:
$$
K = -\frac{1}{2} W
$$
so that its total energy $E = T+W = -T/2 < 0$, and the planet is bound to the star. 
But after the explosion, while the planet speed is left unchanged, its potential energy $W = -GM_\star M_{planet}/r$ is reduced because $M_\star$ has decreased by at least a factor of $6$: the new potential energy $-W_{final} < -W_{initial}/6$. Hence the new energy
$$
E = T + W_{final} = -\frac{1}{2}W_{initial} + W_{final} > -\frac{1}{3} W_{initial} > 0\;:
$$
the final , total energy is positive, the planet is unbound from the star, it will just fly away from it. 
A: A star can be so big that its collapse to a black hole permits no supernova, hence no mass is lost in this way even locally. Some mass/energy is lost by a gravitational wave which may 10-20% of the available mass/energy. This may allow some planets escape from the system. The local amplitude of the gravitational wave might do far more damage to planets than the supernova. More stuff for science fiction.
The distant observer, us with our instruments and LIGO, are in for a much more interesting time. Science fiction is boring compared to science fact.
