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I have the below table:

CREATE TABLE `_loans` (
 `loan_id` int(11) NOT NULL,
 `price` float(7,2) NOT NULL,
 `term` int(11) NOT NULL,
 `app_date` date NOT NULL
) ENGINE=InnoDB DEFAULT CHARSET=latin1;

INSERT INTO `_loans` (`loan_id`, `price`, `term`, `app_date`) VALUES
(1, 299.00, 24, '2019-01-23'),
(2, 774.00, 24, '2019-01-24'),
(3, 817.80, 24, '2019-01-24'),
(4, 279.99, 24, '2019-01-28'),
(5, 463.99, 24, '2019-01-28'),
(6, 0.00, 24, '2019-01-28'),
(7, 357.00, 24, '2019-02-02'),
(8, 386.98, 24, '2019-02-04'),
(9, 846.00, 24, '2019-02-04'),
(10, 713.99, 24, '2019-02-06'),
(11, 0.00, 24, '2019-02-07'),
(12, 579.00, 24, '2019-02-11'),
(13, 179.00, 24, '2019-02-13'),
(14, 0.00, 24, '2019-02-19'),
(15, 259.00, 24, '2019-02-21'),
(16, 249.99, 24, '2019-02-26'),
(17, 319.00, 24, '2019-03-02'),
(18, 1108.99, 24, '2019-03-05'),
(19, 319.00, 24, '2019-03-05'),
(20, 199.97, 24, '2019-03-06');

ALTER TABLE `_loans`
 ADD PRIMARY KEY (`loan_id`),
 ADD KEY `app_date` (`app_date`);

And the below query, which gives a monthly summary of the table data:

SELECT 
MONTHNAME(w.app_date) month,
YEAR(w.app_date) year,
COUNT(*) contracts,
SUM(w.total_price) totals,
@running_total := @running_total + SUM(w.total_price) running_totals
FROM (
 SELECT 
 app_date,
 SUM(price * term) total_price
 FROM _loans l
 GROUP BY l.loan_id
) w
JOIN (SELECT @running_total := 0) r
GROUP BY YEAR(w.app_date), MONTH(w.app_date) 
ORDER BY YEAR(w.app_date), MONTH(w.app_date) ASC

I need a running total of the values from the 'totals' column. Everything works well except the cumulative total, which is not accumulating.

month | year | contracts | totals | running_totals 
------------------------------------------------------------------
January | 2019 | 6 | 63234.72 | 63234.71923828125
February | 2019 | 10 | 85703.04 | 85703.04016113281
March | 2019 | 4 | 46727.04 | 46727.039794921875

This isn't working because of the GROUPing. You need to nest the GROUPing as a subquery and compute the running total from that:

SELECT monthname as month, year, contracts, totals,
 @running_total := @running_total + totals AS running_totals
FROM (SELECT 
 MONTHNAME(w.app_date) monthname,
 MONTH(w.app_date) monthnum,
 YEAR(w.app_date) year,
 COUNT(*) contracts,
 SUM(w.total_price) totals
 FROM (SELECT 
 app_date,
 SUM(price * term) total_price
 FROM _loans l
 GROUP BY l.loan_id
 ) w
 GROUP BY year, monthnum, monthname
) t
JOIN (SELECT @running_total := 0) r
ORDER BY year, monthnum ASC

Output:

month year contracts totals running_totals
January 2019 6 63234.72 63234.71923828125
February 2019 10 85703.04 148937.75939941406
March 2019 4 46727.04 195664.79919433594

Demo on dbfiddle

Your query is overly complicated. The subquery is not necessary for the aggregation. However, it is necessary for the use of variables:

SELECT month, year, total_price,
 (@running_total := @running_total + total_price) as running_totals
FROM (SELECT MONTHNAME(l.app_date) as month,
 YEAR(l.app_date) as year,
 MONTH(l.app_date) as mon,
 COUNT(DISTINCT l.loan_id) as contracts,
 SUM(l.price * l.term) total_price
 FROM _loans l
 GROUP BY l.loan_id, YEAR(w.app_date), MONTH(w.app_date)
 ORDER BY l.loan_id, YEAR(w.app_date), MONTH(w.app_date)
 ) l CROSS JOIN
 (SELECT @running_total := 0) params
ORDER BY year, mon ASC