Hi,
First of all Congratulations.
In UK by rate we mean how much u charge per project or per hour or per day. I am not quite sure about the US industry.
I am pretty sure they wanted to mean if you want to keep yourself free for them. Between that date they will get the rest of the materials which you will need to carry on with the work. Though they are saying will start next week but i might take you to start late-est 29th .
Thank you