If my Mom gave my sister $16,000 to "help them from losing their home" how does that affect her being approved for Medicaid?

Yes, this will be considered a gift. And the amount above the gifting tax law is taxable on her tax return.

Why did she give them $16,000 to pay for two months on a house they knew they were going to lose? It must really be an expensive house for $16,000 would pay for 16 months for my house.

$16,000 probably would have paid for her to be in a nursing home for two months while applying for Medicaid. Now family members will likely need to pay $16,000 on her care before she will qualify for Medicaid.

Oh, I should have looked at your profile first. I see that your mother is already in a nursing home. I still think the family will need to pay $16,000 before she will be approved for Medicaid.