How do I extract a sub-hash from a hash?

If you specifically want the method to return the extracted elements but h1 to remain the same:

h1 = {:a => :A, :b => :B, :c => :C, :d => :D}
h2 = h1.select {|key, value| [:b, :d, :e, :f].include?(key) } # => {:b=>:B, :d=>:D} 
h1 = Hash[h1.to_a - h2.to_a] # => {:a=>:A, :c=>:C} 

And if you want to patch that into the Hash class:

class Hash
  def extract_subhash(*extract)
    h2 = self.select{|key, value| extract.include?(key) }
    self.delete_if {|key, value| extract.include?(key) }
    h2
  end
end

If you just want to remove the specified elements from the hash, that is much easier using delete_if.

h1 = {:a => :A, :b => :B, :c => :C, :d => :D}
h1.delete_if {|key, value| [:b, :d, :e, :f].include?(key) } # => {:a=>:A, :c=>:C} 
h1  # => {:a=>:A, :c=>:C} 

ActiveSupport, at least since 2.3.8, provides four convenient methods: #slice, #except and their destructive counterparts: #slice! and #except!. They were mentioned in other answers, but to sum them in one place:

x = {a: 1, b: 2, c: 3, d: 4}
# => {:a=>1, :b=>2, :c=>3, :d=>4}

x.slice(:a, :b)
# => {:a=>1, :b=>2}

x
# => {:a=>1, :b=>2, :c=>3, :d=>4}

x.except(:a, :b)
# => {:c=>3, :d=>4}

x
# => {:a=>1, :b=>2, :c=>3, :d=>4}

Note the return values of the bang methods. They will not only tailor existing hash but also return removed (not kept) entries. The Hash#except! suits best the example given in the question:

x = {a: 1, b: 2, c: 3, d: 4}
# => {:a=>1, :b=>2, :c=>3, :d=>4}

x.except!(:c, :d)
# => {:a=>1, :b=>2}

x
# => {:a=>1, :b=>2}

ActiveSupport does not require whole Rails, is pretty lightweight. In fact, a lot of non-rails gems depend on it, so most probably you already have it in Gemfile.lock. No need to extend Hash class on your own.