Hi Beth, an interesting question, for which there may be relatively simple answer: As you know, the basic Chinese sentence structure is S V O (subject verb object). As you can see, in your two examples, the structure is S O V, where the object is placed before the verb. The pair of words 把 ...給 allows you to move the object before the verb to give the object emphasis that a normal S V O structure would not give to the object. In other words, 我忘掉了开会的事 is the normal sentence structure, but to emphasize that it is the MEETING that you forgot, you use the S + 把O給 + V structure to move the object before the verb. The result is 我把开会的事给忘掉了。 Hope this is helpful!

我把开会的事给忘了 and 我把开会的事忘了. These 2 expression are both correct. Having “给” there or not does not affect the whole meaning of your expression. But adding "给” is a more colloquial expression in a emphatic tone.

As you may have noticed, it is weird to have 给 in plain descriptions. If someone tries to recall what he has done earlier today, he probably will not use 给. 我把我的中文给忘在家了 in response of "What happened to your Chinese?" sounds pretty natural, and so does the exclamation 我把这些字都给写错了. 给 places emphasis not only on surprise and excitement, it could be embarrassment, sense of regret, or anything. Excluding 给 is more impersonal, and to some extent more formal. 我娘把它给烧了 simply carries much more emotion than 我娘把它烧了.

Now imagine you're watching a play. 阿猫：...然后他就走到那个人面前... 阿狗：怎么啦？ 阿猫：他把裤子(给)脱了！ 阿狗：哟！！然后呢？ 阿猫：把钱给拿出来了！ In the last line, if you pull 给 out, it'll lame the dramatic tension. Consequently, 给 is more likely to stay there, unless you wish to achieve another effect.

I "actually" forgot it somewhere else. He's "literally" blown up.