Ideally you would be using M-values rather than beta values, as they are more amenable to analysis using tools like lmFit. In which case they range from -Inf to Inf (hypothetically), and it wouldn't be surprising at all to have a beta larger than 1 or less than zero.
But your question has more to do with how bumphunter works in a statistical sense, which is ideally something you would understand before using the software. You would be well served to read the bumphunter vignette as well as the papers that describe the method.
I remember bumphunter use coefficient or t-statistic to calculate DMRs, not beta value. Thus the value you can see in the output is not beta value I think, it's a smoothed coef.
And you are right that seems it's pretty hard to find clear explanations for bumphunter output. When I code ChAMP, I basically read every line of bumphunter to understand how it works (because users ask me via email, I am maintaining ChAMP), below is my explanation:
1) value: Mean value of all smooth coef in one candidate bump.
2) L: Numbers of CpGs contained in candidate bump.
3) p.value: Proportion of random bumps show most CpGs and higher mean value then this DMR.
4) fwer: Proportion that a random run would generate one such DMR shows most CpGs and higher mean value.
5) p.valueArea: Proportion of random bumps show higher abs sum value then this DMR.
6) fwerArea: Proportion that a random run would generate one such DMR show higher abs sum value.