The basic argument for a higher minimum wage causing inflation is that when minimum wages rise, other wages rise as well. Companies lose so much in paying their employees that they have to raise prices and fire employees to remain profitable, and as the prices of gas and materials go up, the prices of goods and services using those go up, with everything balancing out in the end.
There are a few flaws with this argument, however. Firstly, while anyone making less than the new minimum gets their wages raised, wages don't usually go up very quickly for those making more than the new minimum already. In particular, wages of CEO's, shareholders, and other executives don't go up immediately in response to minimum wage laws. So the immediate effect on the cost of labor goes mostly to businesses that rely on documented low-wage workers, such as restaurants and retail. The restaurants may raise their prices a bit and the retailers may try to do the same work with less people, but the effect isn't huge. The oil industry tends to have higher paid workers, and the agricultural industry tends to use undocumented workers, so the cost of food doesn't rise quickly as a result.
With more people able to afford goods and services, the economy grows. If current businesses are unwilling or unable to keep up with demand, new businesses come in. This is why unemployment doesn't see a huge uptick after minimum wage increases. Because all the rhetoric given about large corporations being "job-creators" is backwards. Businesses respond to demand, one way or another. Consumers are the real job-creators, because they are the ones actually paying for the result of the labor being done. Also, as people build up wealth, they are more likely to start their own businesses, and to negotiate for better wages. This does then lead to inflation, but it is inflation due to growth, which is good in the short term.
However, this is only a temporary fix, since growth can't continue forever. What we really need is a guaranteed basic income and a shortened work week so that we can have a society that moves toward opportunity equality while shrinking to or maintaining a sustainable economy. But that's a topic for another post.