Why does gabapentin make me more depressed and give me anxiety?

Some experience what you are reporting while others do not experience the same side effects. This is due to numerous factors like the dosage, the frequency, pre-existing conditions, metabolic rate and neurotransmitter levels and overall chemistry. Talk to your prescriber about adjusting the dose then see how you do. Quite often the drug has to be adjusted to YOUR comfort level. Meaning what YOU can tolerate.

Hi, Tessa!
Depression and new/increased anxiety are common side effects that people may experience when first starting on gabapentin:
"Psychiatric
Common (1% to 10%): Abnormal thinking, depression, hostility, emotional lability, insomnia, anxiety, nervousness"
They normally resolve within days to a week or two and they generally lessen in intensity over a period of days.
I'd suggest that you report them to your health care professional for advice before stopping the medication.
Best regards and you should be feeling better quickly now.