Drainage holes are all about allowing water to drain out. Plastic pots with holes at the bottom need to be inside the main outer, decorative pot if there is no drainage holes in the outer pot.
If you don't have drainage holes, the soil can become waterlogged, creating more chance of root rot, even for the most low-maintenance plants for indoor gardening. This is because the roots will only drink the water they require - so any excess water will just sit at the base of the soil. You really need the holes to make sure the soil loses this excess water. Another trick to help drainage along is to put broken-up ceramic or small shells into the bottom of your plant pot, creating gaps and pockets of air where the water can drain down.
For the best results, I would recommend bottom watering your houseplants. The drainage holes will allow the water to get sucked up from the bottom up, going straight to the root rather than having to filter down through various layers of soil.