Translation:Let's hope that she did not start crying.
Speriamo can also be the imperative tense (besides also the indicative and the subjunctive), which translates well as "Let's hope", since the imperative tense is used for instructions and commands etc.
It could also be interpreted as the indicative 'We hope' however, and the only way to tell the difference would be with tone of voice or context, so 'we hope' should also have been correct.
What it shouldn't be interpreted as, however, is the subjunctive mood, which is only used in the dependant clause (the bit that follows che, come etc) in sentences like this.
I think that 'let's hope' is also a reasonable translation. http://blogs.transparent.com/italian/using-the-subjunctive-in-italian/
I do not think that there is a way to tell the difference between 'let's hope' and 'we hope'.
No matter how I try, when I listen to this sentence being spoken by DL all I hear is 'messa piangere' and not 'messa a piangere'. I find this happens a lot. It is not what you can actually hear, but what article you need to know should be there, whether you can hear it or not.