You have corrupted sprites. Pull the PCB from the cabinet. For every socketed chip on the board, carefully lift the chip out without stressing or bending its legs. Look for signs of dust, damage or corrosion on the chip, the chip's legs and the socket. If found, remedy it, then continue. Carefully re-insert the chip and move to the next. The order and position of the chips matters so don't just pull them all at once, or at least without making notes of where exactly they went.
Look over the PCB and look for other signs of damage; check around the capacitors (the little canister looking components on the board) and look for any kind of leakage. Leaky, bulging or exploding capacitors are operating outside of specification and need to be replaced with a new, identical component. Electrolytic fluid is corrosive, but if found early, can simply be wiped up with isopropyl alcohol; check the surrounding area for damage to traces, and repair if connectivity is broken.
If you aren't comfortable working with soldering tools, bring a six pack and make friends with someone who is, or at least get your skills up by disassembling and removing/reinstalling components from some other electronic device you don't care about before you risk damaging your arcade stuff!
The ROMs themselves may also be corrupt; if you have access to a EEPROM programmer, you can compare the data in the ROMs against the ROMs used by MAME. Note there may be multiple versions of the game, so you may need to compare a few variations of the same ROM against your set just to determine which version of the game you have. If you do find a bad ROM, you can replace it using any blanked compatible EEPROM, programming it with the ROM from the MAME set, and then simply reinstalling it in the PCB.
Beyond that, you are looking at some other faulty component upstream of the ROMs. That's going to get outside of the beginner guidance I can provide here -- make local friends who are in the hobby who can help, and be sure to compensate them for their effort.
Hope this helps! Welcome to the hobby!