As Fiona has said, this was long before the days when people were expected to prove everything with bits of paper. It was assumed that people told the truth. Moreover, the Church of England (and thus, by implication, the state) had long had a system for verifying the bona fides of couples planning to marry - the purpose of banns was to spread the word in advance that a couple planned to wed and encourage the local community to speak up if there was an impediment such as an existing spouse, a too close relationship between the parties or the minority of one or both of them.
As the bride in question called herself a widow rather than a spinster, she presumably had children to account for. It is possible that her previous husband had disappeared off the scene and the vicar was prepared to remarry her on the basis that her husband could be "presumed dead". This is a legal fudge whereby a person whose spouse had deserted them utterly and of whom no word had been heard for 7 years could remarry.. It was, subtly, not the same as the Church (or state) actually declaring the person dead, but it was considered to be a watertight defence (for both the bride/groom and clergyman) against a charge of bigamy.
Alternatively, as Fiona has suggested, they just married somewhere distant from home and kept their fingers crossed!