Translation:I am proud of my son.
It's difficult to translate this to English because 자랑스럽습니다 is more of an adjective than a verb, but if you think of this way:
사랑하다 - verb - to love
사랑스럽다 - adjective - to be lovable, i.e. worthy of being loved
자랑하다 - verb - to boast
자랑스럽다 - adjective - to be "boastable" i.e. worthy of being boasted about
I typed "We are proud of our son" 4 months after you posted that. It is still marked wrong, even though the Korean does not specify "I" or "We" or "He" to tell who is proud of a son. "He is proud of his son" is accepted. I reported that my answer should also be accepted. Oct. 14, 2019.
• Adjectives ending in -스럽다 act as causative verbs (~ cause smt to happen) and are intransitive (no object complement, no 을/를).
• In the use of such verbs, the direct object (represented by 을/를) is turned to a complementary object (represented by the complementary marker 이/가)
• 자랑-스럽다 is such verb. = (complementary subject) makes sb proud
(저/우리) 아들이 자랑스럽습니다 - My/our son makes me/us proud. [Causative form]
In Active form,
I/we are proud of my/our son - 아들을 자랑스러워-합니다.
(~을/를) 자랑스러워하다 (Transitive vb) - be proud of, take pride in.