Unfortunately, there is currently no easy way to conveniently export transaction data of your Safe. It is planned for the future, however short term there are 2 options:
(1) Use a third-party service
All Safe transactions are stored on the Ethereum blockchain. Theoretically, any third-party service would be able to access them. The caveat however is, Safe transactions are so-called internal transactions, which are not supported by all third-party accounting solutions.
(2) Fetch the data in JSON from the Safe transaction service
The Safe transaction service indexes all Safe transactions.
There is an endpoint that can be used to return those transactions in a JSON format:
Make sure to use a checksummed Safe address in the URL. For Safe address 0xA063Cb7CFd8E57c30c788A0572CBbf2129ae56B6, the URL would be https://safe-transaction-mainnet.safe.global/api/v1/safes/0xA063Cb7CFd8E57c30c788A0572CBbf2129ae56B6/multisig-transactions/