You can download the latest version of multibit classic from here. How you transfer the Bitcoins to the new wallet depends on which type of backups you made,
https://multibit.org/releases/multibit-classic/multibit-classic-0.5.19/If you have backed up files with a .key extension (that contain exported private keys) you can open multibit, click tools, then click import private keys, then browse to your backed up key file.
If you have backed up multibit wallet files you can open multibit, click file, then click open wallet, then browse to your backed up wallet file.
editAs alani123 says you could alternatively choose to install the electrum wallet and import your keys into that. The maximum fee the latest version of multibit allows is sometimes not enough for fast confirmations if the network's overloaded with unconfirmed transactions. Electrum lets you choose any fee you want, it has no maximum allowed. If you choose electrum download it from the official website
here because there are some fake electrum wallet downloads available that will steal your coins.
If you export your multibit wallet's private key to a file without a password, then open that file in notepad you should see something like the text in the quote below. The bit in blue is the private key. You can import that into electrum using
these instructions.
# KEEP YOUR PRIVATE KEYS SAFE !
# Anyone who can read this file can spend your bitcoin.
#
# Format:
# <Base58 encoded private key>[<whitespace>[<key createdAt>]]
#
# The Base58 encoded private keys are the same format as
# produced by the Satoshi client/ sipa dumpprivkey utility.
#
# Key createdAt is in UTC format as specified by ISO 8601
# e.g: 2011-12-31T16:42:00Z . The century, 'T' and 'Z' are mandatory
#
L4ciWWbEdt3hauU5Tudn13RuN9ZdTqqtm7X1DHbCDgUzpMvQiwsU 2009-01-03T18:15:05Z
# End of private keys