Does that mean the estimated rent of my home will be included in GDP?
Yes.
For example, let's say I bought a new house to live in. The rental fee for a similar house near my place is 2000 dollars per month. Although I'm not paying any rent for my own home, GDP will consider that I consume 2000 dollars every month. Is it correct?
It won't necessarily be calculated this way. The statistical office could also use survey where they would ask you how much would you be willing to pay for your own house. There are various different ways how the rental value of the house you own can be calculated. You need to check meta data and manuals of statistical office that calculates GDP in country you are interested in.
BEA in US for example imputes the rent from similar rental units, but again this is not only way how to do it.
Furthermore, I would like to understand why such a thing is counted in GDP. As we know, GDP doesn't count values generated outside the market, like housekeeping activities from a family member. Is it an attempt to include values generated within a family?
To make GDP invariant to shifts between tenant occupancy and owner occupancy.
Usually the amount of housework people do is more or less constant, so although it is omitted from GDP you are omitting constant and changes in GDP are still measured more or less accurately. Omitting home ownership would create additional bias as there are periodic shifts between raise in home ownership and renting.
In addition, omission of non-market production is generally considered bad aspect of GDP. There are various attempts to actually include estimates of home production into GDP (e.g. see here), although that is problematic due to difficulty of measuring home production. Calculating imputed rent on home is far less controversial.
