Please explain FFT Frequency Spectrum and different filters
I will try and explain things as simply as I can.

The frequency spectrum shows how the filter looks in the frequency domain.That filter is the one which is multiplied (in the frequency domain) with your input signal so you can tell what type of filter your filter is by looking at the frequencies you filter stops e.g a filter with all zeros will stop everything and one with all ones will let all frequencies pass.

For example in you image of the plot of \$H_1[k]\$, look at the frequencies greater than 0 (ignore the negative part of the graph for now - it is just a reflection of the positive side), from the graph you can see that low frequencies in the filter are stopped and higher frequencies are let through, therefore that is a high pass filter.

In the second image (again ignoring the negative parts), you can see that very low frequencies and very high frequencies are let through and only a small band of frequencies is stopped.That filter is therefore a band-stop filter.