▲你的这个句中的 be out 确实不太好理解，所以查了一下此句的出处。在 The Washington Post（华盛顿邮报）查到了，原句的上下文是：

From the beginning of the review, it irked Obama that Petraeus, Mullen and Gen. Stanley A. McChrystal, then the top U.S. commander in Afghanistan, had been out campaigning for more troops on top of the 21,000 that Obama had approved shortly after taking office.

▲be out doing sth 的基本意思是“在外做某事”，其中的doing sth 为伴随状语。如：

He’s out walking the dog. 他出去遛狗去了。

Dan had been out drinking with his friends. 丹和朋友出去喝酒了。

She hasn’t been out riding since the accident. 她自从出了事故以后，一直没有骑马外出过。

有时可引申指“到处做某事”“四处做某事”。如：

I was out beating the bushes for investors to split the risk. 我四处寻找投资者来分担风险。（摘自《新牛津英语双解大词典》）

所以句中的had been out campaigning for… 可理解为“四处呼吁做某事”。

▲campaign 用作动词时，是不及物的，用于campaign for sth / to do sth，其本意是“参加或领导一战役或运动”，在不同语境中可理解为“试图做某事”“呼吁做某事”“为做某事而奔走游说”等。又如：

They campaigned for nuclear disarmament. 他们发起了要求核裁军的运动。

They have campaigned strongly for compulsory registration of dogs. 他们强烈呼吁实行犬类强制登记。

They actively campaigned to save their local cinema. 他们为挽救当地的电影院而积极奔走游说。

▲句中的on top of 在此表示“除……之外”。如：

He writes for the newspaper on top of his regular job. 他除日常工作外，还为报纸写稿。

She receives a substantial housing allowance on top of her salary. 除工资外她还有为数不少的住房津贴。

He has a large private income on top of what he earns as a teacher. 除了做教师的工资之外，他还有一大笔丰厚的私人收入。

▲句中的that Obama had approved shortly after taking office 是定语从句，修饰 the 21000，意思是“奥巴马上任不久就批准的21000人（的部队）”。

▲句子意思是：From the beginning of the review, it irked Obama that Petraeus, Mullen and Gen. Stanley A. McChrystal, then the top U.S. commander in Afghanistan, had been out campaigning for more troops on top of the 21,000 that Obama had approved shortly after taking office. 自策略检讨会开始，奥巴马就感到恼火，因为除了奥巴马上任不久批准的21000人之外，Petraeus, Mullen 和 原美国驻阿富汗最高指挥官Stanley A. McChrystal将军曾四处呼吁，要求派更多的部队。

句首的 review 之所以翻译成“策略检讨会”，是因为这篇报道在前面部分提到了奥巴马要召开第八次策略检讨会（its eighth strategy review session）。