It is real grease. For some reason, silicone doesn't get clean in the dishwasher. I suspect that this is connected to the fact that cleaning methods which work for hard surfaces will not always look for soft ones (see e.g. the research on ultrasound washing machines), but this is just speculation on my part.
What I do is to put the silicone cookware through the washing machine. It emerges without any greasy feeling, it is like rubber on touch. I have been doing this for years and it hasn't harmed anything. I normally put it in in together with the kitchen and bathroom towels, using a long, high-tumble, 60 degrees or higher cycle. Obviously, the temperature doesn't do anything (these moulds go into the oven at 200+ degrees Celsius, the washing machine only goes up to 90 and using this temp is the exception), and the tumbling and centrifugation also seems to cause no problems. I don't put them through a dryer afterwards, just shake off any clinging water.
I only wash mats and soft moulds in the washing machine. I wash utensils with hard plastic parts (spatulas, some of my novelty ice cube trays) per hand in the sink, as I avoid putting hard items in the washing machine if I can, and also don't know how the plastic will react to high temperatures.