First, you can't trust any high speed measurement done on a breadboard, especially with the scope probe having the long ground wire with alligator clip attached. The ground wire not only adds inductance which causes ringing and artifacts, it also makes a 1-turn transformer secondary with the primary being anything in the vicinity. You can experiment by clipping the ground on the probe tip, which turns it into a loop antenna, and hovering it above your circuit while it switches, without actually touching anything with the probe. See what you get on the scope display: in theory, you should get a flat line...
Now the scope traces:
When turning the MOSFET on, the driver injects current into the gate to rise gate voltage. However, the MOSFET has substantial capacitance between G and D. Therefore, some of that gate pulse goes through Cgd and couples into the drain. This results in an upward drain voltage spike at turn-on, which is visible on your "Rising edge" shots. This is normal.
When turning the MOSFET off, gate voltage goes in the other direction, so the same occurs, and you get a downward voltage spike on the drain. This is also normal, and it is visible on both "Falling edge" scope shots, although the vertical scale is different which makes it less conspicuous on the left, but it's there.
If the load is inductive, then the energy stored in that inductor will have to dissipate. The usual fix is to add a freewheeling diode to the supply rail, which should switch fast, so a Schottky diode. Otherwise, the stored energy will end up as heat, either due to damped oscillations, the MOSFET limiting voltage via avalanche, or the MOSFET catching it by switching slowly. This energy is lost on each cycle, which adds losses:
\$ P_{lossL} = F_{req} . \frac{1}{2} L I² = 25mW \$ for 10µH inductance and 0.5A current, so a 0.4% efficiency penalty for driving a 6W LED. Not really an issue.
If you add a cap aross the FET D-S then the energy stored in the inductor will end up charging the cap. On the "Falling edge" scope shot we can see some oscillations as the cable inductance and the 100nF cap ring. The oscillations are damped so the energy is lost. If you don't want to have damped oscillations you can use a R-C snubber instead. Then the 100nF cap charges to about 3V, because it sits between the LEDs and ground.
On the next turn-on the MOSFET will short the cap, and dissipate as heat all the energy stored in it.
\$ P_{lossC} = F_{req} . \frac{1}{2} C (V_1-V_2)² = 9mW \$ or 0.15% efficiency penalty. Okay, that's not catastrophic either.
A more pressing issue is that if you drive LEDs at the end of a long cable, and your driver can switch the MOSFET in 10ns, then that'll generously spread harmonics all over the spectrum over hundreds of MHz and end up making a wideband radio jammer. Try using an AM or FM radio in the vicinity, tuned to a multiple of your PWM frequency.
The fix for that is to switch the MOSFET slowly, 1µs or so. For this kind of application, I use a 74HCT buffer chip as the MOSFET driver. It translates ESP32's 3V3 output to 5V which is more convenient to drive a MOSFET, and its output current is very wimpy which limits MOSFET switching to a more civilized speed, say 1µs or so. If you use a 5V MCU you can drive the FET directly with a PWM GPIO and a gate resistor, say 1kOhm. Observe switching speed on the scope and adjust accordingly.
