恐（おそ）らく is an adverb which originally stems from the conjugation of the yodan verb, 恐（おそ）る ‘to fear’. In most cases, 恐（おそ）らく is simply used as a formal way to say ‘probably’, or ‘perhaps’, and is usually partnered with other structures of conjecture such as だろう, かもしれない, etc.
As 恐（おそ）らく behaves adverbially, it will most often appear at the beginning of sentences before a statement of conjecture is made.

• 妻（つま）は今夜（こんや）会社（かいしゃ）の人（ひと）たちと飲（の）み会（かい）に行（い）くと言（い）っていたので、おそらく11時（じ）ごろぐらいまでは帰（かえ）ってこないと思（おも）います。
  My wife said she was going to a drinking party tonight with some people from work, so she probably won't be home until around 11:00 or so.
• あいつの不自然（ふしぜん）な行動（こうどう）からして、おそらくあいつが犯人（はんにん）だろう！
  Judging by that guy's unnatural behavior, he's probably the culprit!
• おそらく明日（あした）は雨（あめ）かもしれないので、明日（あひた）予定（よてい）していたピクニックは中止（ちゅうし）しましょう。
  Perhaps it may rain tomorrow, so let's cancel the picnic we had planned for tomorrow.

Although 恐（おそ）らく can, and often does, carry the meaning of ‘I fear that (A)’, it isn’t necessarily always used in relation to negative things. In the case of this particular grammar pattern, the main nuance that ‘I fear that (A)’ conveys is modesty on behalf of the speaker in that they fear their assumption may be right, given the available information.